LECTURE 42

FOURIER SERIES PART I

Suppose that a function f has period T = 2pi. Then f may be approximated by the Fourierseries

f(x) = a0 +n=1

(ancosnx+ b

nsin nx)

where the Fourier coefficients a0, an and bn are given by

a0 =1

2pi

pipi

f(x) dx

an

=1

pi

pipi

f(x) cosnx dx (n = 1, 2, 3, . . . )

bn

=1

pi

pipi

f(x) sinnx dx (n = 1, 2, 3, . . . )

.

Suppose that we are dealing with a periodic function f of period 2pi. Periodic functionsare complex in their own special way. We saw earlier that the infinite repetition madethe calculation of their Laplace transforms quite a drama. By using Fourier series wecan bust such functions up as a sum of much simpler periodic trigonometric componentssin(nx) and cos(nx). This is of enormous value when dealing with both differential andpartial differential equations.

In this lecture we will start off with the simpler case of functions with period T = 2pi.Later on the theory will be generalised to functions of arbitrary period T .

We begin with the periodic function

f(x) =

x, 0 x < pi;pi, pi x < 2pi.f(x+ 2pi) otherwise

This is (by definition) is a function of period 2pi and the graph over 6pi x 6pi lookslike

3

2

0

2.5

1.5

151050-10

0.5

1

-5-15

1

The graphs of sin x and cos x are also of period T = 2pi:

1

0

0.5

-0.5

-1

x

1050-5 15-10-15

1

0

0.5

-0.5

-1

x

15105-15 -10 0-5

It is fairly clear that there is no way that f could possibly be approximated by takinglinear combinations of sin(x) and cos(x)....... f looks nothing at all like these two functions!But we have a trick up our sleeves! We can make the trig functions busier (by reducingtheir periods). Consider the graphs of sin 3x and cos 3x:

1

0

0.5

15

-0.5x

100

-1

-15 -10 -5 5

1

0

0.5

15

-0.5x

100

-1

-15 -10 -5 5

(Note that in general the period of sinnx and cos nx is T =2pi

n).

What we then do to create a Fourier series is to use all possible functions sinnx andcosnx together with a constant term a0 to approximate f as an infinite series:

f(x) = a0 +n=1

(ancosnx+ b

nsinnx)

2

The coefficients a0, an and bn are referred to as the Fourier coefficients. The bad newsis that their calculation is usually a tedious process involving the integral formulae

a0 =1

2pi

pipi

f(x) dx

an

=1

pi

pipi

f(x) cosnx dx (n = 1, 2, 3, . . . )

bn

=1

pi

pipi

f(x) sinnx dx (n = 1, 2, 3, . . . )

.

We will prove these formulae in the next lecture but for today lets look at what needsto be done to calculate the Fourier series of our function.

Example 1 Find the Fourier series of the function f above on page 1.

Some equations (amongst others) that you must have at your fingertips for Fourier seriesare:

cosnx dx =

1

nsinnx+ C

sin nx dx =

1

ncosnx+ C

sinnpi = 0

cosnpi = (1)n

sin(x) = sin x

cos(x) = cos x

sin 0 = 0

cos 0 = 1

3

4

a0 =3pi

4an=

(1)n 1

n2pibn=1

n

f(x) =3pi

4+

n=1

{(1)n 1

n2pi

}cosnx+

{1

n

}sinnx

f(x) =3pi

4+2

picosx sin x

1

2sin 2x

2

9picos 3x

1

3sin 3x+ . . .

Observe that both anand b

ntend to zero as n. This always happens under normal

circumstances and guarantees that the Fourier series will converge for standard functions.

5

It is fascinating to look at how the Fourier series steps up in accuracy as the numberof terms increases. The three graphs below show the first partial sum

3pi

4+2

picosx sin x

the third partial sum

3pi

4+2

picosx sin x

1

2sin 2x

2

9picos 3x

1

3sin 3x

and finally the tenth partial sum.

3.5

2.5

0.5

3

2

15-5

11.5

50-150

10-10

3.5

2.5

0.5

3

2

1510

11.5

-5 50

-10-15 0

3

2

0

2.5

1.5

1510

0.51

-15 5-5 0-10

If we were to take infinitely many terms then the Fourier series would sit right over theoriginal function. Observe that the nth partial sums are continuous and yet they do a finejob of approximating discontinuous objects. We will examine carefully at a later stagewhat actually happens at the points of discontinuity, a situation known as the Gibbsphenomenon .

It is fair to say that the construction of a Fourier series is a major undertaking, butthe payoffs are huge. In the next lecture we will still restrict our attention to functionswith a period of 2pi but will develop some shortcuts which will help us out on occasion.

42You can now do Q 108

6